Abstract

In the context of nonparametric density estimation, we consider the combination of automatic bandwidth selection rules with qualitative constraints on the search space of bandwidths derived from bounds on the number of modes. These constraints can be easily combined with an upper bound based on the concept of oversmoothing introduced by Terrell and Scott (1985). Rather obviously, if a correct upper bound on the number of modes is known, our proposed approach helps to ensure an adequate representation of known qualitative features by the estimate. More surprisingly, even loose upper bounds on the number of modes are able to improve the MISE behavior of least squares cross-validation, by reducing the known tendency of under-smoothing of this bandwidth selector.